To solve the equation \( 9e^{5t} = 11e^{4t} \), follow these steps:

1. **Divide both sides by \( e^{4t} \):**
   \[
   \frac{9e^{5t}}{e^{4t}} = \frac{11e^{4t}}{e^{4t}}
   \]

   Simplifying the left-hand side:
   \[
   9e^{5t - 4t} = 11
   \]

   This simplifies to:
   \[
   9e^t = 11
   \]

2. **Isolate \( e^t \):**
   \[
   e^t = \frac{11}{9}
   \]

3. **Take the natural logarithm of both sides to solve for \( t \):**
   \[
   \ln(e^t) = \ln\left(\frac{11}{9}\right)
   \]

   Using the property \( \ln(e^t) = t \):
   \[
   t = \ln\left(\frac{11}{9}\right)
   \]

4. **Express \( t \) more simply:**
   \[
   t = \ln(11) - \ln(9)
   \]

   This is the exact solution for \( t \). You can also approximate it numerically if needed:
   \[
   t \approx 0.207
   \]

Would you like further details or have any questions? 

Here are some related questions you might be interested in:
1. How do you solve exponential equations with different bases?
2. What are the properties of natural logarithms?
3. How do you solve logarithmic equations?
4. How can exponential functions be used to model real-world scenarios?
5. What is the inverse of an exponential function?

**Tip:** Always use logarithmic properties to simplify expressions when solving equations involving exponents.

Solve the exponential equation 9e^5t = 11e^4t by simplifying it and applying logarithmic properties. This step-by-step solution explains how to isolate the variable and find the natural logarithm, making it suitable for students in Grades 10-12.

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